We will give a positive answer in Section .
The main idea in our mind is to generalize the phase space analysis (WKB analysis associated with adoptable diagonalization procedure) from the Euclidean space \(\mathbb {R}^n\)  to the Heisenberg group \(\mathbf {H}_n\) . Despite the fact that so far it seems no any research on the Cauchy problem for linear coupled system on the Heisenberg group from the knowledge of the author, motivated by the recent papers [1]}, [2]}, we will apply the group Fourier transform on \(\mathbf {H}_n\) . In order to overcome the difficulty from the phenomenon that the group Fourier transform of a \(L^2(\mathbf {H}_n)\)  function is no longer a function rather a family of bounded linear operator, we can work with the Hermit functions. Nonetheless, on the Heisenberg group, differently from
